My teaching career began as a high school mathematics teacher, yet my focus over the last ten years has been in elementary, both researching and now teaching. I am currently teaching math to two Grade 5/6 classes, with half of them having IPPs. They are a complex group.
My interest in elementary math began by watching my own kids struggle. What was holding them back? Why did some students struggle more than others? Both of my kids, early in school, were labelled with learning disabilities in mathematics. Now both are achieving at a high level, one in high school, the other in junior high. My work with them was a journey that ebbed and flowed between barriers and progress. It often felt like going through a maze, heading in one direction, and then hitting a barrier. So, we would turn around and try another direction. Over time, patterns emerged in those barriers, and one telltale characteristic began to reveal itself: memorization. Through this recognition, the barriers became easier to avoid. Upon hitting a barrier, I would ask, “OK, what in this task am I expecting them to just remember without understanding?” Once identified, we would go back and focus on developing the conceptual understanding or image in their mind for this idea or symbol. Once they were no longer expected to memorize a process or a symbol without understanding, they would progress in leaps and bounds, often exceeding my expectations. Where there had previously been a brick wall there was now a passageway.
The barrier for many students is not the math but the ability to remember. Presenting students with symbols along with a series of steps that represent concepts before they have sufficiently grown their own personal understanding or image for that concept can be a major barrier. Students who memorize easily have an advantage, but is that advantage rooted in mathematical understanding? I have worked with many students who can find an answer but do not understand the underlying math.
How can we be more inclusive while focusing on the growth of understanding for the majority? It was this question that took me down the path of exploring how we absorb information and what types of activities contribute to the growth of our mental images for mathematical ideas. Are there ways of offering students information that are richer than others?
Different approaches to offering information
May I suggest three categories for offering information, ranked according to their ability to give us information as directly, originally, and optimally as possible:
- The lowest and most empty way is the signitive, or linguistic, form of mathematics.
- An indirect offering of information is the imaginative – the utilization of our own mental images.
- The most direct way is the perceptual, as the acting out of mathematical ideas.
The first mode of presenting a mathematical idea is through the oral discussion of mathematics and its written symbols. This is the emptiest way of presenting the meaning within a mathematical idea. We absorb information through our senses, and these symbols visually look nothing like the ideas they represent. If a student has not grown the understanding of these symbols, we are not offering them anything with meaning. Symbols are just the tip of the conceptual iceberg; the meaning, which is so much bigger than the symbol, lies underneath the surface. The symbol for four (4) can represent a distance, position, or quantity, which can be represented in an infinite number of ways. The shape of the symbol (4) itself offers no meaning; it is the students themselves who bring the meaning.
The second form, the imaginative, is the act of visualizing. Zimmerman and Cunningham (1991) state that the intuition that mathematical visualization affords is not a vague kind of intuition; rather, it is what gives depth and meaning to understanding. This internal offering of information for the idea through your personally developed images has much to offer in terms of growth in understanding. Images beget more images, leading to deeper understanding.
It has been well documented in both sports and music that growth occurs through the act of visualizing. As a teenager, my husband’s swim coach would ask the swimmers to lie on deck with their eyes closed while he orally described a swim race, guiding them through it, while they imagined themselves in the race. The description by the coach is a signitive offering, but where each swimmer takes those oral descriptions imaginatively is different for each person. In a meta-analysis done to answer the question Does mental practice enhance performance? (Driskell et al., 1994) the researchers concluded that “mental practice is effective for both cognitive and physical tasks; however, the effect of mental practice is significantly stronger the more a task involves cognitive elements” (p. 485). In this discussion, it is noteworthy to mention that physical practice was the most effective form, and that those who were experienced in physical practice (perceptual) benefited more from mental practice (imaginative) than those who were novices in the task. The reason suggested for this is that, “the novices who mentally practiced a physical task may not have sufficient schematic knowledge about successful task performance and may be spending their effort imagining task behaviors that could turn out to be somewhat counterproductive” (p. 490). This supports the idea of the importance of establishing the perceptual level of activity, which allows for continued growth when visualizing.
Our world is perceived not only in terms of object shapes and spatial relationships, but also in terms of environmental possibilities for action. A perceptual experience in mathematics, I suggest, is the acting out of a mathematical idea. It is within the spatial acting out that meaning is given or “lived” in the most direct, original, and optimal way, for we are experiencing the mathematical idea through our senses when, for example, we physically cut the shape into equal parts, connecting us to the concept of fractions. This physical act will allow for growth when we later imagine this act.
It is important to distinguish and emphasize the importance of this third category (perceptual) because it is a sensory act. It is this act that allows for the deepening and continual growth of images. So, when a student is stuck and struggling to understand, some kind of perceptual experience must be offered, some level of active interaction with the environment to promote further growth.
Understanding the Imaginative and the Perceptual together as Spatial Reasoning
The imaginative and perceptual are closely intertwined. In fact, they are hard to discuss as separate entities, for together they are one idea – spatial reasoning. Spatial reasoning is more than just passively receiving sensations; it is the intentional act of perceiving and then engaging our bodies purposefully (Khan et al., 2015). Through acting out a mathematical idea there is a co-evolving that occurs in both our mental and physical skills. The actions being discussed are not only physical actions, spatial reasoning encapsulates mental actions as well. Visualizing is very productive within mathematics, as spatial reasoning ability and mathematical ability have been shown to be intimately linked (Mix & Cheng, 2012). Mental interactive playing and exercising of our images can stimulate growth in and of themselves without actually engaging our environment, but a foundation for this imagining must first be established (e.g. physically counting, organizing, regrouping, building, drawing, etc.). It is through these physical acts that our imaginative and perceptual experiences interact seamlessly with each other, building and strengthening images.
The idea that the math classroom benefits from the interaction between the signitive, imaginative, and perceptual is the arena in which my research lies. The perceptual and imaginative strengthen and evolve as we engage with our environment, but also within mathematics there is a strong signitive element that must be attached (memorization) to our mental images. How can these elements interact to the mutual benefit of all three?
My research in this area
The classroom teacher and I worked with her Grade 5 students in a school for students labelled with learning disabilities. We began with the foundational ideas of fractions. In our preassessment the students presented as having minimal understanding, many not knowing how to write a fraction (signitive). We had four days with them, offering perceptual experiences that were always combined with the signitive to encourage the association with their growing images. We would have them physically cut shapes (perceptual), practising the concept of splitting into equal parts. Yet before they cut the shapes, we would discuss and imagine how to ensure that they would end up with equal parts (imaginative), i.e. we would fold the shapes and then cut them. Next, the students would write the symbols representing the fraction pieces (signitive). This interaction with pieces offered them a visual, embodied, and imaginative experience connected to the mathematical concept. Later they would combine (perceptual) these different pieces and write the fractions (signitive) using an addition symbol.
Although we started with the basics, we continued stretching the complexity of the topic to see how far their images/understanding would take them. By day four, we played a game called imagine-build-steal, in which we offered them a signitive question first, such as 2/4 + 1/2 The students were then asked to imagine and give a solution. None were able to answer the question based solely on this signitive offering; they had not yet grown sufficient images. They needed more from their environment to deepen their own images. To promote this further enhancement, the students were asked to build (perceptual) using the fraction pieces that they themselves had cut out. This they could do; they had grown sufficient images for looking at the signitive offering and building the solution. So, we continued with this cycle of signitive first, then asking to imagine, and then offering a perceptual experience. Their images continued to grow until by the end of that period, students began to offer up imaginative solutions to expressions like 2/3 + 2/6 + 2/8, based solely on the signitive offering. They had reached the point of sufficient growth of their personal images to solve this complex expression without having to build it.
Making a shift in our classrooms
The growth of mathematical understanding is a complex process, as seen over and over again with my own kids, in my research, and in the classroom. It is also personal; each student must grow their own images for the mathematical ideas in order to be able to visualize and make use of them. For some, they can be slippery. This is true of both the student who finds it easy to memorize and the one who does not. I find spatial reasoning tasks to be a great equalizer in a classroom. The student who struggles to memorize and therefore follow steps may reason and visualize with ease, but those who can follow a series of steps to an answer may struggle to visualize the mathematical concept. Math, however, is about ideas and concepts, not a set of memorized rules. My experience and my research support the claim that mental images are a key element to mathematical understanding that is often underappreciated. Far too frequently, the goal is to get the student as quickly as possible to an answer rather than to deep understanding of the idea.
If these mental images are the key to deep understanding, then what factors influence the growth of these images? If we accept the idea that images are grown through a dynamic process of restructuring based on a stream of perceptual encounters and conceptual revelations (Arnheim, 1969), then playing with, utilizing, and exercising these images can support their growth. A classroom focused on growing images is one in which students are engaged in imagining, drawing, moving, and regrouping objects while incorporating the signitive to encourage association. If all we offer students are symbols on a piece of paper, only those students who have already grown sufficient images can benefit from such a task. As educators, we are then not providing new opportunities for growth to the various levels of student understanding that every classroom contains.
Arnheim, R. (1969). Visual thinking. University of California Press.
Driskell, J., Copper, C., & Moran, A. (1994). Does mental practice enhance performance? Journal of applied psychology, 79(4), 481–492.
Khan, S., Francis, K., & Davis, B. (2015). Accumulation of experience in a vast number of cases: Enactivism as a fit framework for the study of spatial reasoning in mathematics education. ZDM: The International Journal of Mathematics Education, 47(2), 269–279.
Mix, K. S., & Cheng, Y. L. (2012). The relation between space and math: Developmental and educational implications. In J. B. Benson (Ed.), Advances in child development and behavior (Vol. 42, pp. 197–243). Academic Press.
Zimmermann, W., & Cunningham, S. (1991). Editors’ introduction: What is mathematical visualization. In W. Zimmermann, & S. Cunningham, Visualization in teaching and learning mathematics (pp. 1–7). Mathematical Association of America.